TECHNIQUE 4 OF 4
Reverse planning: build the position before the last jump
Working backward clarifies what a finish needs. Choose a possible final hole, then reconstruct the two pegs that could produce it in one legal jump.
The rule and its limit
Immediately before a final jump, the destination is empty and the source and intervening hole each hold a peg. A reverse construction replaces a peg at the destination with those two pegs in empty holes. Repeat only legal reverse constructions; reversing their order then provides a forward solution. A desired final hole is a planning choice, not a requirement of the classic game.
A common mistake
Adding arbitrary pegs around a chosen finish does not create a verified puzzle. The added source and middle must align with the emptied destination in the exact reverse of a legal jump.
PLAYABLE EXAMPLE 1
Read a finish backwards
Imagine undoing the final jump: one peg splits into two with an empty landing left behind. Use that idea to explain a complete forward finish.
Loading playable study…
Select a peg, then one of its available empty landing squares. A jump removes the peg in between.
Keyboard: Tab to an enabled control, then press Enter or Space. The study starts from the diagram shown; reload can restore your own saved moves.
Try the task before inspecting the answer.
Compare the starting moves
Every jump removes exactly one peg, so a one-peg finish needs 6 jumps. Every legal first jump is evaluated. A successful continuation is verified by a complete finishing route; a no-finish conclusion exhausts all remaining legal branches. The route below is one verified finish; its last peg may be anywhere on this classic board. Read the final jump backward: the finish has its only peg at R3C3. Remove that peg and restore pegs at R5C3 and R4C3. That is exactly the two-peg position before the last jump. Continue reversing the written route from bottom to top: each undo restores its source and middle, and empties its landing. These reverse operations explain construction; the playable board permits only forward jumps.
| First move | Immediate effect | Checked consequence |
|---|---|---|
| R3C2 → R1C2 | Remove the jumped peg at R2C2. 6 pegs remain. | No one-peg finish exists after this jump; every remaining legal branch was checked. |
| R3C2 → R5C2 | Remove the jumped peg at R4C2. 6 pegs remain. | No one-peg finish exists after this jump; every remaining legal branch was checked. |
| R4C2 → R4C4 | Remove the jumped peg at R4C3. 6 pegs remain. | No one-peg finish exists after this jump; every remaining legal branch was checked. |
| R4C3 → R4C1 | Remove the jumped peg at R4C2. 6 pegs remain. | Can finish with one peg in 5 further jumps. |
| R5C3 → R3C3 | Remove the jumped peg at R4C3. 6 pegs remain. | No one-peg finish exists after this jump; every remaining legal branch was checked. |
| R5C3 → R5C5 | Remove the jumped peg at R5C4. 6 pegs remain. | No one-peg finish exists after this jump; every remaining legal branch was checked. |
| R5C4 → R5C2 | Remove the jumped peg at R5C3. 6 pegs remain. | Can finish with one peg in 5 further jumps. |
PLAYABLE EXAMPLE 2
Build and unwind a longer chain
Imagine undoing the final jump: one peg splits into two with an empty landing left behind. Use that idea to explain a complete forward finish.
Loading playable study…
Select a peg, then one of its available empty landing squares. A jump removes the peg in between.
Keyboard: Tab to an enabled control, then press Enter or Space. The study starts from the diagram shown; reload can restore your own saved moves.
Try the task before inspecting the answer.
Compare the starting moves
Every jump removes exactly one peg, so a one-peg finish needs 7 jumps. Every legal first jump is evaluated. A successful continuation is verified by a complete finishing route; a no-finish conclusion exhausts all remaining legal branches. The route below is one verified finish; its last peg may be anywhere on this classic board. Read the final jump backward: the finish has its only peg at R3C3. Remove that peg and restore pegs at R3C5 and R3C4. That is exactly the two-peg position before the last jump. Continue reversing the written route from bottom to top: each undo restores its source and middle, and empties its landing. These reverse operations explain construction; the playable board permits only forward jumps.
| First move | Immediate effect | Checked consequence |
|---|---|---|
| R2C2 → R4C2 | Remove the jumped peg at R3C2. 7 pegs remain. | No one-peg finish exists after this jump; every remaining legal branch was checked. |
| R2C3 → R4C3 | Remove the jumped peg at R3C3. 7 pegs remain. | No one-peg finish exists after this jump; every remaining legal branch was checked. |
| R2C3 → R2C1 | Remove the jumped peg at R2C2. 7 pegs remain. | No one-peg finish exists after this jump; every remaining legal branch was checked. |
| R2C5 → R4C5 | Remove the jumped peg at R3C5. 7 pegs remain. | Can finish with one peg in 6 further jumps. |
| R3C2 → R1C2 | Remove the jumped peg at R2C2. 7 pegs remain. | Can finish with one peg in 6 further jumps. |
| R3C2 → R3C4 | Remove the jumped peg at R3C3. 7 pegs remain. | Can finish with one peg in 6 further jumps. |
| R3C3 → R1C3 | Remove the jumped peg at R2C3. 7 pegs remain. | Can finish with one peg in 6 further jumps. |
| R3C3 → R3C1 | Remove the jumped peg at R3C2. 7 pegs remain. | No one-peg finish exists after this jump; every remaining legal branch was checked. |
| R3C5 → R1C5 | Remove the jumped peg at R2C5. 7 pegs remain. | No one-peg finish exists after this jump; every remaining legal branch was checked. |
Adding a spare peg is not a reverse jump
A proposed construction starts with a single peg at R3C3 and adds a peg at R1C1 while leaving R3C3 occupied.
The tempting move
Claim the two-peg result has a guaranteed solution because it was made by working backward from one peg.
Why it fails
No legal reverse jump occurred. The original destination was not emptied, and the two resulting pegs do not form a source-and-middle pair for a jump. They have no legal move at all.
What you can conclude
Use the exact reverse operation, such as emptying R3C3 and placing pegs at R3C1 and R3C2. Record the reversed sequence if you want a constructive solvability proof.
Try the idea in another position
You want the final peg at R5C3. Give two possible two-peg predecessor positions that finish there in one jump, assuming all other holes are empty.
Check your reasoning
Pegs at R5C1 and R5C2 can finish with a jump from R5C1 to R5C3. Pegs at R3C3 and R4C3 can finish with a jump from R3C3 to R5C3. In both predecessor positions R5C3 must initially be empty.